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Why is the order parameter in N=2 Seiberg-Witten theory $\langle \text{tr} \phi^2 \rangle$? (And discussion of gauge-variant order parameter in general)

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In this paper, Seiberg and Witten use the gauge invariant order parameter $\langle \text{tr} \phi^2 \rangle$ to parametrize the breaking of gauge symmetry (I'm using the standard abuse of terminology here, of course gauge symmetry cannot be broken, what's broken is the corresponding global symmetry). But since this is gauge invariant, how can it tell us at all whether the gauge symmetry is broken? Just like in QCD, the flavor-chiral order parameter $\langle \bar{\psi} \psi \rangle$ is QCD-gauge invariant, but it would've been absurd to conclude QCD gauge symmetry is broken just because $\langle \bar{\psi} \psi \rangle\neq 0$. (In fact it puzzles me why some people take gauge invariance of the order parameter as a virtue instead of a deficiency in this case.)

A late edit: much of the discussion (which is enlightening to me) has been centered around aboout "does $\langle \phi \rangle$ make sense at strong coupling?", but this is actually somewhat a diversion of what I wanted to say. The thing is I'm not even convinced $\langle \phi^2 \rangle\approx\langle\phi\rangle^2$ at weak coupling region. For example, a weakly coupled Ising model is in its disordered phase, if we use $\phi$ to denote lattice spin, then $\langle\phi\rangle=0$, while $\phi^2=\text{Id}$ which is nonzero in any possible phase.

Anther point raised by 40227 is Elithur's theorem, however the theorem only applies in a gauge-invariant quantization scheme, such as lattice gauge theory, while unfortunately Seiberg and Witten never explicitly specify what kind of quantization scheme they are having in mind.

@40227, it doesn't have to have a direct meaning, although at the very least it tells us if the vacuum condensate is "charged". Nonzero vev of a gauge-variant field gives non-trivial phenomenology, like Higgs mechanism.

In fact a more physical example would be field-theoretic treatment of superconductivity. In this case you want to know if the vacuum is electrically charged, so you have to look at the vev of a charged field, which must be gauge variant.

Can you give me a reference to relativistic superconductivity, so that I can investigate the matter?

It seems to me that having a solid around already breaks the Poincare-symmetry down to a 3D lattice symmetry $\times$ time translations, which would completely change the situation compared to what has been discussed in algebraic QFT. Therefore, at present I have no intuition of what might happen for such a symmetry group.

@ArnoldNeumaier, I think color-superconductivity would count. Although the formalism treating it is often not manifestly covariant (which is not surprising since e.g. it's often dealt with in thermal QFT context), it's still a QCD phenomenon after all.

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To understand what is going on one has to make a difference between what is a full/quantum/non-perturbative quantum field theory and what is a Lagrangian and/or semiclassical/perturbative description of a theory. In a full QFT, one has an algebra $\mathcal{A}$ of (physical) fields of operators. A (global) symmetry group $G$ is a group of automorphisms of this algebra. A choice of vacua is a choice of realization of $\mathcal{A}$ on an Hilbert space $H$, space of (physical) states (the states in $H$ are obtained from a particular vector in $H$, the vacuum, by action of elements of $\mathcal{A}$). We can have different choices of vacua corresponding to different (inequivalent) representations of $\mathcal{A}$ on an Hilbert space. For a given choice of vacua, all the symmetries of $\mathcal{A}$ are not necessarely realizable by unitary transformation of the Hilbert space: the realizable symmetries form a subgroup $H$ of $G$ and if $H$ is strictly smaller that $H$, one has spontaneously symmetry breaking from $G$ to $H$. The spectrum of the theory in the given vacua contains one massless scalar (Goldstone boson) for each continuous direction in $G/H$.

The notion of gauge symmetry depends on a specific Lagrangian description of the theory. In such description, one starts with a classical field theory with a gauge symmetry and one defines a QFT by quantization, let's say by the path integral approach. In such picture only gauge invariant classical fields define corresponding fields of operators in the quantum theory. Indeed to define correlation functions in the quantum theory one has to take the path integral over gauge equivalence classes of fields and so only gauge invariants quantities can be included in the integrand. One could try to define correlation functions of gauge variants fields by fixing a gauge and it is indeed possible perturbatively but the results depend on the gauge choice and fixing a gauge is anyway in general impossible at the non-perturbative level (Gribov ambiguity). So, very concretely, in the Seiberg-Witten example, $\phi$, which is a gauge-variant field in the classical starting point of the Lagrangian description, does not define a well-defined field of operators in the full QFT and in particular it does not make sense to talk about an expectation value $<\phi>$.

In the classical theory, it makes sense to say that the field $\phi$ has a non-zero value at infinity, the usual description of the Higgs mechanism applies and this story extends to the perturbative level. To understand the relation with the full non-perturbative theory, it is useful to think in terms of path integrals. A Lagrangian for a gauge theory defines a full QFT by path integral over gauge equivalence classes of classical fields. In particular, one has the choice of boundary conditions at infinity for the classical fields we are integrating over and this choice is mapped to the choice of vacuum of the full quantum QFT. But this mapping can be quite non-trivial. In the Seiberg-Witten story, the boundary condition on the field $\phi$ is specified by a complex number $a$, well-defined up to a sign. Classically, the moduli space of classical vacua is parametrized by $a$. For $a \neq 0$, the gauge symmetry is spontaneously broken from $SU(2)$ to $U(1)$ and for $a=0$ the $SU(2)$ gauge symmetry is unbroken. For big $a$, the classical theory is weakly coupled at the symmetry breaking scale and so one expects that for every such $a$ the path integral with boundary conditions prescribed by $a$ defines a vacuum of the full quantum theory, with an infrared behaviour looking like the classical one: a U(1) gauge theory with massive W bosons. But for small $a$, the classical theory is strongly coupled and it is unlikely that the quantum theory looks like the classical one. In fact the path integral has infrared divergences making the correspondence between $a$ and quantum vacua doubtful. The conclusion is that $a$, what would be a candidate for $<\phi>$, is not a good well defined coordinate on the moduli space of vacua. It is not very surprising precisely because $\phi$ is not an allowed observable in the full theory.

Breaking of gauge symmetry is not breaking of a corresponding global symmetry simply because in general there is no global symmetry associated to a gauge symmetry. More precisely the conserved current associated to a global gauge transformation is in general gauge variant and so cannot define a well defined charge on the Hilbert space of (physical) states. (A well known exception to this statement is QED where the current associated to the global $U(1)$ is gauge invariant and there is a well defined eletric charge but to have a spontaneous symmetry breaking one needs a charged scalar and the current associated to global $U(1)$ is not gauge invariant because of the term $A^\mu A^\nu \phi \phi^\dagger$ in the Lagrangian). If there were really a breaking of a global symmetry then one should see a Goldstone boson.

The conclusion is that the notion of spontaneous symmetry breaking of a gauge symmetry only makes sense given a Lagrangian/ classical/ perturbative description of the theory. It is not surprising as gauge symmetry is simply a redundancy in a given description of the theory (digression: physical consequences of a gauge symmetry description exist at the level of asymptotic symmetries but they are much more sublter objects that a global symmetry acting on the Hilbert space). So asking the question: is there a spontaneously symmetry broken of a gauge symmetry in a given vacuum of a full non-perturbative QFT does not really make sense. A question which makes sense is: are there some massless spin 1 particles ? If yes then there is a natural gauge theory description. If no then it's no.

So the meaningful questions that Seiberg and Witten are trying to answer are: what is the space of vacua of the theory and what is the infrared physics in each of these vacua? They start by the classical story, with a moduli space parametrized by $a$, a $U(1)$ unbroken gauge symmetry at $a\neq 0$ and a $SU(2)$ unbroken gauge symmetry at $a =0$. They argue that this picture is qualitatively correct at the quantum level for large $a$. To study the general case, one needs a good coordinate on the space of vacua. Natural functions on the space of vacua are vev of fields of operators. $tr \phi^2$ is a well-defined field of operator of the theory because it comes from a gauge invariant function in the path integral definition of the QFT and so it makes sense to consider $<tr \phi^2>$. The fact that it is a good choice is not obvious a priori, it could be a constant function on the space of vacua for example. But it is a good choice because it is clearly a good choice in the region where the classical approximation is good, for large $a$, $<tr \phi^2> \sim a^2$. In other words, $<tr \phi^2>$ is the simplest way to extend to the full quantum theory the variable $a$ natural from the classical point of view. All the work is then to determine the quantum corrections to the classical picture, and in particular to compute exactly $<tr \phi^2>$ as a function of $a$ in the region of the space of vacua where $a$ is still a good coordinate.

@ Jia Yiyang : the point of my answer is that the notion of "spontaneously broken vacua" does not make sense away from the perturbative regime. In the perturbative regime, the classical story is a correct approximation and in the classical strory $tr \phi^2 \neq 0$ implies $\phi \neq 0$, hence spontaneously broken gauge symmetry. In QCD, the chiral condensate is already a non-trivial dynamical non-perturbative effect and at this level I don't know what gauge symmetry breaking means. But certainly I agree with you that, in a classical story, a non-trivial expectation value of a gauge invariant field does not imply in general a spontaneous gauge symmetry breaking. Sometimes, as for $tr \phi^2$, non-trivial value of a gauge invariant field is related to a non-trivial value of a gauge variant field and so is a signal for gauge symmetry breaking. Sometimes it is not.

I'd say over all their ranges. Or may I answer you this way: Over the same range that makes you think 〈trϕ2〉 is well defiend?

If you take all the range then you probably obtain an infinite result because along the gauge orbits the action is constant and so there is no exponential suppression of the integrand.

To give a path integral definition of a gauge invariant quantity like $<tr \phi^2>$, one has to integrate over the quotient space by the gauge transformations, i.e. on the gauge equivalence classes of fields. This makes sense because a gauge invariant quantity naturally defines a function on this space whereas it is not the case for a gauge variant quantity.

Let me give an elementary finite dimensional analogue: take the real line $\mathbb{R}$ as analogue of the space of fields, the additive group $\mathbb{Z}$ as analogue of the group of gauge transformations, acting on $\mathbb{R}$ by translation. The quotient space $\mathbb{R}/\mathbb{Z}$ is a circle $S^1$ and is the analogue of the space of gauge equivalence classes of fields. The analogue of a gauge invariant quantity is a function on $\mathbb{R}$ invariant under integral translations, i.e. a 1-periodic function. The analogue of a general gauge variant quantity is a general function on $\mathbb{R}$. It is clear what is the mean value of a 1-periodic function: it is the integral over any interval of length $1$. It is unclear what is the mean value of a general function: the integral over the real line will in general diverge.

maybe I should ask a question that's more relevant to the title. How critical is the usage of 〈trϕ2〉〈trϕ2〉 in Seiberg-Witten paper? Had they used 〈ϕ〉〈ϕ〉, does it go terribly wrong?

Again the point is that we don't know what $<\phi>$ means. In the Seiberg-Witten context, one could naively think that there is a way to define $<\phi>$. Indeed, classically, $<\phi>$ makes sense, is determined up to gauge transformations by its eigenvalues $(a,-a)$, and according to the classical Higgs mechanism, $|a|$ is (maybe up to some numerical constants) the mass of the $W$ bosons. This suggests a definition of $|a|$ in the full quantum theory: define it as the mass of the $W$ bosons. This works in the semiclassical region of the moduli space of vacua where there exists $W$ bosons but this fails in the strong coupling region: it is unclear a priori what is the spectrum of the theory and if there is in the spectrum something one could call $W$ bosons (and in fact it is part of the conclusion of the Seiberg-Witten analysis that the $W$ bosons existing at weak coupling disappear at strong coupling). One can try to do better: classically $a$ is the electric component of the central charge. The central charge is part of the supersymmetry algebra and so makes sense in the full quantum theory (we are looking at supersymmetric vacua). This suggests a definition of $a$ in the full quantum theory: define it as the electric part of the central charge. But this fails because the "electric part" is not well-defined because of the electromagnetic duality in 4d abelian gauge theory. More precisely, it is well-defined in the weak coupling region and one can "analytically continue" it in the strong coupling region but due to singularities in the moduli space of vacua, one can come back in the weak coupling region with a non-trivial monodromy. So $a$ is a multivalued function on the moduli space of vacua and so is not enough to describe the moduli space. Conversely, $u=<Tr \phi^2>$ is a good coordinate on the moduli space and the subject of the Seiberg-Witten paper is to determine the relation between $u$ and $a$: for a given vacuum $u$, what is the central charge, what is the spectrum...

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